4. [Among Professor Boole's manuscripts I found five pages in German, forming part of a memoir, which was probably intended for Crelle's Mathematical Journal. The memoir was to have discussed two applications of the Calcu lus of Variations; one to the Jacobian Theory of the Last Multiplier, and the other to the Solution of Pfaff's equation X để + X để + 2 2 2n But there is only a single paragraph relating to the second application. The manuscript contains the same demonstration of the Jacobian Theory of the Last Multiplier as in Art. 2 of the present Chapter; after this demonstration some remarks occur of which the substance will now be given.] It is worthy of notice, that Jacobi in the 36th volume of Crelle's Journal, deduced by the aid of the Calculus of Variations the result on which the preceding demonstration of the Theory of the Last Multiplier depends. In fact, he shewed that if V denotes any function of and V be transformed by the introduction of a new system of independent variables u,, u,, ... u, then the following relation holds, Jacobi applies this result to the transformation of the expression But neither Jacobi himself, nor any other person, so far as I know, has drawn attention to the application of the result which I have given here. [The substance of the single paragraph relating to the second application of the Calculus of Variations will now be given.] Clebsch has earned the thanks of all who are interested in the higher parts of the Theory of Differential Equations, since he has performed the same service for Pfaff's problem as Jacobi did for the Theory of Partial Differential Equations of the first order, and thereby for the equations of Dynamics. But while I recognise the great importance of the results, I consider it desirable to give a simpler deduction of the system of partial differential equations therein involved, and on which the other results depend. CHAPTER XXXII. THE DIFFERENTIAL EQUATIONS OF DYNAMICS. [IT will be seen that this is only a fragment of the Chapter which was to have appeared under this title.] I do not propose in this Chapter to discuss the origin and interpretation of the differential equations of motion or to enter into those details of their application which are found in all ordinary treatises on Dynamics. But they constitute a system analytically so remarkable from the forms in which it is capable of being expressed, and from the general methods of integration which emerge out of those forms, that they are well deserving of a special attention. Referred to rectangular co-ordinates the differential equations for the motion of a system of points free or connected |